The characteristic mapping method uses a computational framework for non-linear advection capable of resolving fine scale fluid phenomena without the necessity of increasing the resolution of the computational grid. By approximating the inverse flow map generated by a velocity field as a composition of submaps, the method generates a discretization with an exponentially increasing polynomial degree of approximation using only a linear increase in the degrees of freedom. This functional spatio-temporal discretization has the capacity of accurately and sparsely representing fine scales globally, substituting the effects of spatial refinement with the operation of composition. As a step towards the application of these techniques to geophysical fluid phenomena, we present a characteristic mapping method for the rotating barotropic vorticity equations. The method is verified using standard test cases demonstrating third-order accuracy in the supremum norm. Numerical experiments illustrating the ability to reproduce the direct energy cascade at finer scales than the computational grid are provided.

翻译：特征映射方法采用一种非线性平流的计算框架，能够在无需提高计算网格分辨率的情况下解析精细尺度的流体现象。该方法通过将速度场生成的逆流映射近似为子映射的复合，利用自由度仅线性增加的方式，生成具有指数递增多项式近似阶数的离散化。这种时空离散化方案能以稀疏且精确的方式全局表征精细尺度，将空间细化的效果替换为复合运算。为推进此类技术在地球物理流体现象中的应用，我们提出了一种适用于旋转正压涡度方程的特征映射方法。通过标准测试案例验证，该方法在最大模范数下具有三阶精度。数值实验表明，该方法能够以比计算网格更精细的尺度再现直接能量级联的特征。